Greetings,

I am trying to build a setup where I have a document set content type and a list content type inside the document set content type. So that users are given a form to fill out when they create the document set and then when they create a document inside the document set, another form is presented to them to fill out.

Is this possible? If not, what are the alternatives?

If you want users to fill out a form when they upload a document inside a docSet, why don't you create a library content type, and then allow that content Type inside the docSet? you don't need a list content type...

on library settings -> docset -> Document Set Settings -> allowed content types, you can choose which content types will be available inside that docSet